Explicit points on the Legendre curve III

Abstract

We continue our study of the Legendre elliptic curve $y^2=x(x+1)(x+t)$ over function fields $K_d={\mathbf{F}}_p({\mu }_d,t^{1/d})$. When $d=p^f+1$, we have previously exhibited explicit points generating a subgroup $V_d$ of $E(K_d)$ of rank $d-2$ and of finite, $p$-power index. We also proved the finiteness of $III(E/K_d)$ and a class number formula: $[E(K_d):V_d{]}^2=|III(E/K_d)|$. In this paper, we compute $E(K_d)/V_d$ and $III(E/K_d)$ explicitly as modules over ${\mathbf{Z}}_p[\mathrm{Gal}(K_d/F_p(t))]$.

v3: Section added to correct an error in an intermediate result. Main results not affected

Cited by in corpus (1)

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